TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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A left or right exact functor between abelian categories is automatically additive

Statement

Let $F : A \to B$ be a functor between abelian categories. If $F$ is left exact or right exact, then $F$ is additive.

Facts & Assumptions

Given: A functor $F : A \to B$ between abelian categories.

[L1]

Left exact means preserving finite limits, and right exact means preserving finite colimits (Left exact and right exact functors).

[L2]

Abelian categories are additive (Abelian category).

[L3]

A functor between additive categories is additive exactly when it preserves finite biproducts (A functor between additive categories is additive exactly when it preserves finite biproducts).

Proof

technique · direct

1.1L1L2L3

If $F$ is left exact, then by [L1] it preserves the zero object and binary products. In an additive category, products are biproducts by [L2], so $F$ preserves finite biproducts.

1.2L1L2L3

If $F$ is right exact, then by [L1] it preserves the zero object and binary coproducts. Again [L2] identifies those with finite biproducts, so $F$ preserves finite biproducts.

2.1L3step 1.1step 1.2∎

In either case, [L3] applies and shows that $F$ is additive.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Section 12.7, Lemma 12.7.2(1)